Jacob Burbea

Abstract

In this paper we investigate the
relationship between total positivity and reproducing kernels. We extend the notion
of total positivity to domains in the complex plane. In doing so, we also give a
geometrical interpretation to certain Wronskians of reproducing kernels. These
geometrical quantities are connected to Gaussian curvatures of Kähler metrics
induced by these kernels. For simply-connected domains these curvatures are negative
constants, thereby showing that the kernels are totally positive and moreover
yielding an efficient method for computing the relevant determinants. In
general, the reproducing kernels of multiply-connected domains are not totally
positive.